Quantum Speedups of Optimizing Approximately Convex Functions with Applications to Logarithmic Regret Stochastic Convex Bandits
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We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set K⊆ℝ^n and a function Fℝ^n→ℝ such that there exists a convex function f𝒦→ℝ satisfying sup_x∈ K|F(x)-f(x)|≤ϵ/n, our quantum algorithm finds an x^*∈ K such that F(x^*)-min_x∈ K F(x)≤ϵ using Õ(n^3) quantum evaluation queries to F. This achieves a polynomial quantum speedup compared to the best-known classical algorithms. As an application, we give a quantum algorithm for zeroth-order stochastic convex bandits with Õ(n^5log^2 T) regret, an exponential speedup in T compared to the classical Ω(√(T)) lower bound. Technically, we achieve quantum speedup in n by exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in T for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.
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